Olympiad problems

2011 Olympic Revenge, 5

Let $n \in \mathbb{N}$ and $z \in \mathbb{C}^{*}$. Prove that $\left | n\textrm{e}^{z} - \sum_{j=1}^{n}\left (1+\frac{z}{j^2}\right )^{j^2}\right | < \frac{1}{3}\textrm{e}^{|z|}\left (\frac{\pi|z|}{2}\right)^2$.

2009 Indonesia TST, 4

Tags: inequalities , inequalities proposed

Let $ a$, $ b$, and $ c$ be positive real numbers such that $ ab + bc + ca = 3$. Prove the inequality \[ 3 + \sum_{\mathrm{\cyc}} (a - b)^2 \ge \frac {a + b^2c^2}{b + c} + \frac {b + c^2a^2}{c + a} + \frac {c + a^2b^2}{a + b} \ge 3. \]

2013 JBMO TST - Turkey, 5

Tags: inequalities proposed , inequalities

Let $a, b, c ,d$ be real numbers greater than $1$ and $x, y$ be real numbers such that \[ a^x+b^y = (a^2+b^2)^x \quad \text{and} \quad c^x+d^y = 2^y(cd)^{y/2} \] Prove that $x<y$.

2021 Kyiv City MO Round 1, 11.4

Tags: inequalities , algebra , inequalities proposed

For positive real numbers $a, b, c$ with sum $\frac{3}{2}$, find the smallest possible value of the following expression: $$\frac{a^3}{bc} + \frac{b^3}{ca} + \frac{c^3}{ab} + \frac{1}{abc}$$ [i]Proposed by Serhii Torba[/i]

1999 Irish Math Olympiad, 1

Tags: inequalities proposed , inequalities

Find all real numbers $ x$ which satisfy: $ \frac{x^2}{(x\plus{}1\minus{}\sqrt{x\plus{}1})^2}<\frac{x^2\plus{}3x\plus{}18}{(x\plus{}1)^2}.$

2005 APMO, 2

Tags: inequalities , algebra , inequalities proposed , APMO

Let $a, b, c$ be positive real numbers such that $abc=8$. Prove that \[ \frac{a^2}{\sqrt{(1+a^3)(1+b^3)}} +\frac{b^2}{\sqrt{(1+b^3)(1+c^3)}} +\frac{c^2}{\sqrt{(1+c^3)(1+a^3)}} \geq \frac{4}{3} \]

2008 Bulgaria National Olympiad, 2

Tags: induction , inequalities proposed , inequalities

Let $n$ be a fixed natural number. Find all natural numbers $ m$ for which \[\frac{1}{a^n}+\frac{1}{b^n}\ge a^m+b^m\] is satisfied for every two positive numbers $ a$ and $ b$ with sum equal to $2$.

2011 Vietnam National Olympiad, 1

Tags: inequalities , inequalities proposed

Prove that if $x>0$ and $n\in\mathbb N,$ then we have \[\frac{x^n(x^{n+1}+1)}{x^n+1}\leq\left(\frac {x+1}{2}\right)^{2n+1}.\]

2012 Junior Balkan Team Selection Tests - Moldova, 2

Tags: inequalities , inequalities proposed

Let $ a,b,c $ be positive real numbers, prove the inequality: $ (a+b+c)^2+ab+bc+ac\geq 6\sqrt{abc(a+b+c)} $

2024 Baltic Way, 5

Tags: inequalities , inequalities proposed , algebra , algebra proposed

Find all positive real numbers $\lambda$ such that every sequence $a_1, a_2, \ldots$ of positive real numbers satisfying \[ a_{n+1}=\lambda\cdot\frac{a_1+a_2+\ldots+a_n}{n} \] for all $n\geq 2024^{2024}$ is bounded. [i]Remark:[/i] A sequence $a_1,a_2,\ldots$ of positive real numbers is \emph{bounded} if there exists a real number $M$ such that $a_i<M$ for all $i=1,2,\ldots$

2004 Baltic Way, 3

Tags: inequalities , inequalities proposed

Let $p, q, r$ be positive real numbers and $n$ a natural number. Show that if $pqr = 1$, then \[ \frac{1}{p^n+q^n+1} + \frac{1}{q^n+r^n+1} + \frac{1}{r^n+p^n+1} \leq 1. \]

2013 Mediterranean Mathematics Olympiad, 3

Tags: inequalities , inequalities proposed

Let $x,y,z$ be positive reals for which: $\sum (xy)^{2}=6xyz$ Prove that: $\sum \sqrt{\frac{x}{x+yz}}\geq \sqrt{3}$.

2011 Indonesia MO, 7

Tags: inequalities proposed , inequalities

Let $a,b,c \in \mathbb{R}^+$ and $abc = 1$ such that $a^{2011} + b^{2011} + c^{2011} < \dfrac{1}{a^{2011}} + \dfrac{1}{b^{2011}} + \dfrac{1}{c^{2011}}$. Prove that $a + b + c < \dfrac{1}{a} + \dfrac{1}{b} + \dfrac{1}{c}$.

2005 Georgia Team Selection Test, 10

Tags: inequalities , inequalities proposed , 3-variable inequality , algebra

Let $ a,b,c$ be positive numbers, satisfying $ abc\geq 1$. Prove that \[ a^{3} \plus{} b^{3} \plus{} c^{3} \geq ab \plus{} bc \plus{} ca.\]

2007 Bulgaria Team Selection Test, 3

Tags: inequalities proposed , inequalities

Let $n\geq 2$ is positive integer. Find the best constant $C(n)$ such that \[\sum_{i=1}^{n}x_{i}\geq C(n)\sum_{1\leq j<i\leq n}(2x_{i}x_{j}+\sqrt{x_{i}x_{j}})\] is true for all real numbers $x_{i}\in(0,1),i=1,...,n$ for which $(1-x_{i})(1-x_{j})\geq\frac{1}{4},1\leq j<i \leq n.$

2015 China Team Selection Test, 1

Tags: inequalities , inequalities proposed , China TST

Let $x_1,x_2,\cdots,x_n$ $(n\geq2)$ be a non-decreasing monotonous sequence of positive numbers such that $x_1,\frac{x_2}{2},\cdots,\frac{x_n}{n}$ is a non-increasing monotonous sequence .Prove that \[ \frac{\sum_{i=1}^{n} x_i }{n\left (\prod_{i=1}^{n}x_i \right )^{\frac{1}{n}}}\le \frac{n+1}{2\sqrt[n]{n!}}\]

2007 All-Russian Olympiad Regional Round, 11.8

Tags: inequalities , induction , inequalities proposed

Prove that $ \prod_{i\equal{}1}^{n}(1\plus{}x_{1}\plus{}x_{2}\plus{}...\plus{}x_{i})\geq\sqrt{(n\plus{}1)^{n\plus{}1}x_{1}x_{2}...x_{n}}\forall x_{1},...,x_{n}> 0$.

2006 Poland - Second Round, 3

Tags: inequalities , inequalities proposed

Positive reals $a,b,c$ satisfy $ab+bc+ca=abc$. Prove that: $\frac{a^4+b^4}{ab(a^3+b^3)} + \frac{b^4+c^4}{bc(b^3+c^3)}+\frac{c^4+a^4}{ca(c^3+a^3)} \geq 1$

2012 Junior Balkan Team Selection Tests - Moldova, 2

Tags: floor function , inequalities proposed , inequalities

Let $ a,b,c,d$ be positive real numbers and $cd=1$. Prove that there exists a positive integer $n$ such that $ab\leq n^2\leq (a+c)(b+d)$

1987 Romania Team Selection Test, 9

Tags: trigonometry , inequalities proposed , inequalities

Prove that for all real numbers $\alpha_1,\alpha_2,\ldots,\alpha_n$ we have \[ \sum_{i=1}^n \sum_{j=1}^n ij \cos {(\alpha_i - \alpha_j )} \geq 0. \] [i]Octavian Stanasila[/i]

2009 South East Mathematical Olympiad, 3

Tags: inequalities , inequalities proposed

Let $x,y,z $ be positive reals such that $\sqrt{a}=x(y-z)^2$, $\sqrt{b}=y(z-x)^2$ and $\sqrt{c}=z(x-y)^2$. Prove that \[a^2+b^2+c^2 \geq 2(ab+bc+ca)\]

2010 Contests, 1

Tags: inequalities , rearrangement inequality , inequalities proposed

Let $a_1,a_2,\cdots, a_n$ and $b_1,b_2,\cdots, b_n$ be two permutations of the numbers $1,2,\cdots, n$. Show that \[\sum_{i=1}^n i(n+1-i) \le \sum_{i=1}^n a_ib_i \le \sum_{i=1}^n i^2\]

1983 IMO Longlists, 49

Tags: inequalities , inequalities proposed

Given positive integers $k,m, n$ with $km \leq n$ and non-negative real numbers $x_1, \ldots , x_k$, prove that \[n \left( \prod_{i=1}^k x_i^m -1 \right) \leq m \sum_{i=1}^k (x_i^n-1).\]

2024 German National Olympiad, 6

Tags: algebra proposed , algebra , inequalities , inequalities proposed , Mean Inequality Chain

Decide whether there exists a largest positive integer $n$ such that the inequality \[\frac{\frac{a^2}{b}+\frac{b^2}{a}}{2} \ge \sqrt[n]{\frac{a^n+b^n}{2}}\] holds for all positive real numbers $a$ and $b$. If such a largest positive integer $n$ exists, determine it.

2003 Hong kong National Olympiad, 1

Tags: LaTeX , inequalities proposed , inequalities

Find the greatest real number $K$ such that for all positive real number $u,v,w$ with $u^{2}>4vw$ we have $(u^{2}-4vw)^{2}>K(2v^{2}-uw)(2w^{2}-uv)$